And now for your amusement and stimulation Vampire Population Ecology Introduction or A Little Math Never Hurt Anyone We are gathered here today to ponder the ways in which the humans and vampires of Sunnydale interact Specifically Betsy asked Ooh Brian can you help us work out the vampire carrying capacity of a typical population I m assuming a typical vampire accounts for say 150 200 humans a year So how big does a town have to be to support Sunnydale s apparently limitless supply of vampires Are there human warrens in the catacombs somewhere used only for feeding purposes The term carrying capacity isn t often applied to predator population dynamics Instead ecologists generally estimate stable predator populations by first coming to grips with the prey s population dynamics including its carrying capacity Actually in a lot of different cases the prey s carrying capacity ultimately determines how well the predator does In principle ecologists might employ two basic strategies to get at a problem like this The empiricists would go out and find a field site where they could actually observe predators and their prey and just tally the results over time The theoreticians would chuckle at the empiricists and construct mathematical models that probably approximate the behavior of populations in the field keeping their hands more or less clean in the process In real life most ecologists use both strategies off and on Unfortunately I don t know of any real life vampire populations in the field so we re going to have to pretend that we are strict theoreticians That means that we ll be using math some algebra some calculus and some matrix theory This is O K It hurts a lot less than say getting bitten by a vampire as you re trying to fit the bugger with a radio collar A Model What follows is based on some of the simpler theoretical understandings of predator prey population dynamics I m assuming that human populations are not controlled solely by vampire predation i e in the absence of vampires the human population would still eventually be limited by some other factor like food supply disease or access to a wellwritten weekly news magazine I like The Economist myself but that s clearly a digression If we let H stand for the size of the human population and V stand for the size of the vampire population then we can represent the changes in each population over time with a pair of differential equations dH rH K H dt K aHV dV baHV mV sV dt where r K a b m s is the intrinsic growth rate of the human population incorporating natural rates of both birth and death as well as immigration is the human carrying capacity of the habitat in question is a coefficient that relates the number of human vampire encounters to the number of actual feedings is the proportion of feedings in which the vampire sires the victim i e this is the vampire birth rate is the net rate of vampire migration into Sunnydale is the rate at which the Scoobies stake vampires assumed to be the only important source of vampire deaths What we need to do simply put is find the equilibria that exist between these two equations In other words we need to find the combination s of human and vampire populations sizes that satisfy both equations at the same time As it turns out there are three such equilibria Without showing you the really ugly math after all this is a family forum I ll just say that two of the equilibria are not very interesting They are 1 when both the humans and the vampires are completely extinct and 2 when the vampires are extinct but the human population hovers at or near its carrying capacity The third equilibrium is the one we care about wherein humans and vampires coexist At that point the solutions are s m r m s H and V 1 ba a baK Notice that the actual human population doesn t depend on carrying capacity at all and that the vampire population does yes only an ecologist could put an exclamation point after a statement like that We know from the existence of our other equilibria that the human population is not necessarily big enough to support a vampire population What we need to know is whether or not the human carrying capacity is large enough Specifically if K is too small then m s 1 baK and the equilibrium vampire population size will be negative Basically the human population s carrying capacity must be higher than its equilibrium abundance K m s ba If this isn t the case then the even largest possible human population isn t large enough and vampires have no hope in this particular region A Trial Now that we have a model we can start trying out some assumptions or if we re lucky actual measurements for the various parameters To start with we know from the sign in Lover s Walk that the human population of Sunnydale is 38 500 We also know that the town of Berkeley CA has a population of about 100 000 Since Berkeley is also a town with a UC campus and is furthermore a town that has been more or less completely urbanized the population has been stable or dropping slightly for about 25 years we will take 100 000 as the carrying capacity for a California university town Let s assume the following Sunnydale s human population growth rate is 10 annually which is at the high end for a budding California community A vampire feeds every three days and encounters about one hundred potential victims in the course of a day meaning that 1 out of every 300 encounters involves a little refreshment An individual vampire sires a victim every other year or once per 240 feedings Buffy and her Slayerettes busy little beavers that they are annually stake about 1 3 of the vampires plaguing Sunnydale Vampires are flocking to Sunnydale since the Hellmouth is the underwordly equivalent of Silicon Valley and the demon labor market is just too good to be true Thus we ll assume a yearly migration rate of about 10 or the same as for the humans In terms of our model we have r 0 0953 a 0 00333 b 0 00417 s 0 600 K 100000 m 0 0953 note that for r s and m I ve pulled a little switcheroo In our assumptions we speculated as to the yearly rates of growth or migration or what have you But our model is based on a set of continuous exponential functions rather than discrete time step geometric functions For our assumptions to make sense in the context of our model I ve had to perform a natural log transformation of the yearly rates For example r ln annual human population growth rate ln 1 10 0 0953 So now we can plug these numbers into our